SymbolicWedderburn.jl

Amazing package providing symbolic but explicit

• decomposition of group representations into (semi-)simple representations and
• the Wedderburn decompositions for endomorphisms of those representations.

We work with

• a (linear) orthogonal actions of a finite group G on finite dimensional vector space V over K = ℝ or K = ℂ (i.e. a KG-module V) and
• linear, G-equivariant maps (equivariant endomorphisms) f : V → V.

These objects are of primary inportance in the study of the representation theory for finite groups, but also naturally arise from (non)commutative polynomial optimization with group symmetry. The aim of the package is to facilitate such uses.

A bit of theory

By Maschke's theorem V can be decomposed uniquely V ≅ V₁ ⊕ ⋯ ⊕ Vᵣ into isotypic/semisimple subspaces Vᵢ and each of Vᵢ ≅ mᵢWᵢ is (in a non-canonical fashion) isomorphic to a direct sum of mᵢ copies of irreducible/simple subspaces Wᵢ. By (symbolic) computation in (the group algebra) KG, SymbolicWedderburn is capable of producing the exact isomorphism V ≅ V₁ ⊕ ⋯ ⊕ Vᵣ in the form of a collection of projections πᵢ : V → Vᵢ either in the group algebra (lazy, unevaluated form), or in terms of projection matrices, when a basis for V is explicitly given.

The isomorphism produces a decomposition of End_G(V) (the set of linear G-equivariant self-maps of V) in the sense of Artin-Wedderburn theorem, i.e. the projections πᵢ block-diagonalize f ≅ f₁ ⊕ ⋯ ⊕ fᵣ where fᵢ : Vᵢ → Vᵢ. In terms of matrices if f is given by n×n-matrix, then we can rewrite it as a block diagonal matrix with blocks of sizes nᵢ×nᵢ where Σᵢ nᵢ = n = dim V and each nᵢ = mᵢ · dim Wᵢ.

Semi-definite constraints

For example, if a basis for a semi-definite constraint admits an action of a finite group, then the semi-definite matrix of a invariant solution is such an equivariant endomorphism. In particular if P is a positive semidefinite constraint, when searching for an G-invariant solution we may replace

0 ⪯ P[1:n, 1:n]

by a sequence of constraints

0 ⪯ P[1:nᵢ, 1:nᵢ] for i = 1…r,

greatly reducing the computational complexity (The size of psd constraint is reduced from n² to Σᵢ nᵢ²). Such replacement can be justified if e.g. the objective is symmetric and the set of linear constraints follows a similar group-symmetric structure.

If we are only interested in the feasibility of an optimization problem, then such replacement is always justified (i.e. an invariant solution is a honest solution which might not attain the same objective).

Rank one projection to simple components

Sometimes even stronger reduction is possible when the acting group G is sufficiently complicated and we have a minimal projection system at our disposal (The package tries to compute such system by a heuristic algorithm. If the approach fails please open an issue!). In such case we can (often) find subsequent projections Vᵢ → K^{mᵢ} (depending only on the multiplicity of the irreducible, not on its dimension!). This leads to an equivalent formulation for the psd constraint with nᵢ = mᵢ further reducing its size.

Moreover in the case of symmetric optimization problems it's possible to use the symmetry to reduce the number of linear constraints (since in that case only one constraint per orbit is needed). SymbolicWedderburn facilitates also this simplification.

Example

In Aut(𝔽₅) has property (T) we use the trick above to successfully simplify and solve a large semidefinite problem coming from sum-of-squares optimization.

The original problem had one (symmetric) psd constraint of size 4641×4641 and 11_154_301 linear constraints. By exploiting its (admittedly -- pretty large) symmetry group (of order 3840) we can reduce this problem to 20 (symmetric) psd constraints of sizes

[56  38  34  32  27  27  23  23  22  22  18  17  9  8  6  2  1  1  1  1]

which correspond to (the simple) Wᵢ blocks above. In particular, the number of variables in psd constraints was reduced from 10_771_761 to just 5_707.

Moreover, the symmetry group has just 7 229 orbits (when acting on the subspace of linear constraints), so the symmetrized problem has equal number of (a bit denser) linear constraints.

The symmetrized problem is solvable in ~20 minutes on an average office laptop (with 16GB of RAM).

For more examples you may have a look at dihedral action example, or different sum of squares formulations.

Related software

Sum of Squares optimization

This package is used by SumOfSquares to perform exactly this reduction, for an example use see its documentation.

The software for sum of (hermitian) squares computations in a non-commutative setting (group algebra of a infinite group) using SymbolicWedderburn is my project PropertyT.jl (unregistered). There we used the sum of squares optimization to prove Property (T) for special automorphisms group of the free group. It's a cool result, check it out!.

Other symbolic decompositions

The main aim of GAP package Wedderga is to

compute the simple components of the Wedderburn decomposition of semisimple group algebras of finite groups over finite fields and over subfields of finite cyclotomic extensions of the rationals.

The focus is thus on symbolic computations and identifying isomorphism type of the simple components. SymbolicWedderburn makes no efforts to compute the types or defining fields, it's primary goal is to compute symbolic/numerical Wedderburn-Artin isomorphism in a form usable for (polynomial) optimization. Wedderga also contains much more sophisticated methods for computing a complete set of orthogonal primitive idempotents (i.e. a minimal projection system) through Shoda pairs. In principle those idempotents could be computed using Oscar and used in SymbolicWedderburn.

Citing this package

If you happen to use SymbolicWedderburn please cite either of

• M. Kaluba, P.W. Nowak and N. Ozawa $Aut(F₅)$ has property (T) 1712.07167, and
• M. Kaluba, D. Kielak and P.W. Nowak On property (T) for $Aut(Fₙ)$ and $SLₙ(Z)$ 1812.03456.

(Follow the arxiv link for proper link to the journal.)